About Many-Valued Logics
This workshop on Many-Valued Logics initiated a two-part series on Non-Classical Logics and Paradox. It is distinctive of Many-Valued (or Fuzzy) Logics that their set of truth values is not restricted to the classical "true" and "false", but rather comprises a larger number of truth degrees. A vague statement such as "Paul is drunk", for example, may accordingly be said to be true to a degree of 70%. Many-Valued Logics can be usefully applied not only in mathematics, linguistics, hardware design, and artificial intelligence, but also inside philosophy. In particular, they can be employed to deal with paradoxes such as the Sorites.
Abstracts and Slides
Vincenzo Marra
Is Truth Gradable? From Bivalence to Many-Valued Logics
In classical logic it is traditional to assume that each sentence can only receive one of two truth values, the verum and the falsum. Common sense would have it, though, that in everyday life the bivalence assumption is crude. Indeed, with the important exception of mathematical discourse, it is hard to produce examples of predicates that are naturally bivalent: most of the time, a predicate will not just apply either fully or else not at all, but rather to a certain "degree", whatever the latter means. And we do use such expressions as "this is much more true than that". Should common sense be put back where it belongs according to E.T. Bell – "on the topmost shelf next to the dusty canister labeled ‘discarded nonsense’" – or is there really a sensible and useful non-classical logic of gradable truth? I will show in this first tutorial how close semantical analysis does indeed lead to appropriate many-valued logics that provide a formal model of sound inferences with ordinary non-bivalent predicates such as ‘Tall’ or ‘Red’. Our analyisis, however, will also tell us that it is wrong to expect the existence of a single tractable formal system for all gradable predicates that fail bivalence. As a consequence, there arises the need for a minimal formal framework within which one can perform appropriate design choices. Such a formal framework is introduced by Petr Cintula in the second part of the tutorial.
Petr Cintula
Mathematical Aspects of Many-Valued Logics
In the second tutorial we showcase the basic mathematical properties of Lukasiewicz and Godel-Dummett logics, two prominent and illustrative examples of logics gradable truth. We introduce their Hilbert-style proof systems and three forms of algebraic semantics: a general one, one based on linearly ordered algebras and finally the so-called standard one built over the real unit interval. Then we present elementary self-contained proofs of three completeness theorems with respect to the three distinct semantics mentioned above. In the process we touch other important logical properties like the formal proof, deduction theorem, proof by cases, finitarity, compactness, etc. We end the tutorial by showing that the presented results can be vastly generalized and actually serve as a basis for a design of a minimal formal framework for working with logics of gradable truth.
Vincenzo Marra
Geometric Aspects of Lukasiewicz Logic
As we have seen in the tutorial, Lukasiewicz logic can be conceived of as the propositional fragment of the logic of vague predicates satisfying a specific set of semantical assumptions; such are, for instance, the predicates ‘Tall’, ‘Fast’, ‘Near’, etc., but not, e.g., ‘Red’ or ‘Cute’. In this talk we look more closely at Lukasiewicz logic with the aim of exhibiting the deep connections that this system entertains with the geometry of polyhedra in Euclidean space. The classical deduction theorem – q follows from p if, and only if, it is the case that p implies q – will be our thread of Ariadne in our exploration of the fascinating semantics of Lukasiewicz logic. The geometric intuition that we will have built upon exiting this mathematical labyrinth is put to good use by Sam van Gool in the last talk of the day.
Sam van Gool
On (Uniform) Interpolation in Non-Classical Logic
Interpolation is a property that was first introduced and proved for classical predicate logic by Craig in the 1950's. In this talk, we will focus on interpolation properties in the context of non-classical propositional logics, in particular in the logics of Godel-Dummett and Lukaciewicz that were discussed by the previous speakers. The purpose and interest in doing so is two-fold. First, Craig's interpolation property and its derivatives often provide non-trivial information about the logic and imply that the logic has several other "good" properties, such as for example the property of Beth definability. Second, the topic of interpolation allows us to illustrate how the algebraic and geometric intuitions from the previous talks may be put to use for obtaining results in logic. Specifically, we will sketch some of the geometric arguments that are involved in proving interpolation properties for both Godel-Dummett and Lukaciewicz logics. We will end by mentioning some current research and open questions related to (uniform) interpolation and predicate logics.